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After some googling, I have found this definition of a parameter in statistics:

"a numerical characteristic of a population, as distinct from a statistic of a sample"

Or, more generally for mathematics:

"a quantity whose value is selected for the particular circumstances and in relation to which other variable quantities may be expressed"

But my understanding of parameters is that they are the betas (coefficients) that appear before the variables (which may be themselves without any transformation or may appear with an exponent, log, square root, or some other transformation,) in a regression equation, and are interpreted as the partial effects of the independent variables on the dependent variable.

A few questions: Are parameters the same as coefficients? If not, what makes them different? While linear regression uses parameters that are linear, how would non-linear parameters look and be interpreted? Are there any concrete examples of parameters in the real world, such as some constant relationship in physics?

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  • $\begingroup$ Please provide sources for your quotes; who said what, where, and in what context? $\endgroup$ – Glen_b -Reinstate Monica Nov 24 '16 at 0:38
  • $\begingroup$ While it doesn't directly address at least some of your questions, you may find some additional insight from each of the answers at Is any quantitative property of the population a “parameter”?. I'd suggest starting with Nick Cox's answer there, but read them in whatever order suits you. $\endgroup$ – Glen_b -Reinstate Monica Nov 24 '16 at 2:17
  • $\begingroup$ Between the thread linked to by @Glen_b, which explains what parameters are and gives general examples, and several threads on linear and nonlinear models (such as stats.stackexchange.com/questions/148638), which provide many examples of parameters in nonlinear regression models, these questions already have extensive answers: please consult them and, if you wish, search our site for more. $\endgroup$ – whuber Nov 24 '16 at 2:57
  • $\begingroup$ @whuber do you think this should close as duplicate? While I didn't see it that way, I don't object to that conclusion. I'm happy to close it if you think so (or you can, either way is fine) $\endgroup$ – Glen_b -Reinstate Monica Nov 24 '16 at 3:09
  • $\begingroup$ @Glen_b I'm on the fence. The request for "real-world" examples of nonlinear regression models is arguably not a duplicate, and it's one aspect your answer addresses explicitly. Many people, however, would view that part of the question as being so lacking in research (there must be hundreds of such examples readily available on CV) that they would argue it should be ignored, leaving the preliminary part a duplicate of the thread you identified. $\endgroup$ – whuber Nov 24 '16 at 4:06
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I'd agree with the first definition in relation to the meaning of the word "parameter" in statistics.

As it says, it's some population characteristic. So for example, a population mean or a population tenth percentile, population maximum, population interquartile range, population correlation, population second Pearson skewness, population coefficient of variation would all be examples of parameters.

Where you have a population distribution specified by some finite set of parameters (such as $\mu$ and $\sigma$ in the normal), these population parameters will be defined in terms of those distribution parameters. So in a normal ($N(\mu,\sigma^2)$) the population interquartile range is about $1.349\sigma$, but in an exponential with rate parameter $\lambda$ the population interquartile range is about $1.0986/\lambda$.

An example of a parameter that's not a regression coefficient can be found even in a regression -- the obvious one would be the population variance of the error term ($\sigma^2$).

Population regression coefficients (of which the sample regression coefficients you obtain by fitting a regression model are intended to be estimates) are certainly examples of parameters, but parameters needn't be the coefficients in a regression. They can be pretty much any numerical characteristic of a population.

Another example of a parameter would be the shift parameter of a shifted exponential distribution; there was a recent question here where a regression model with an additive exponential noise term was being used (where interest focused on estimating a population line which determined the shift parameter -- that is, to estimate the line defining the lower edge of where the data could be).

You can also have parameters for the mean in a nonlinear regression; at least some of them must enter the expectation non-linearly.

An example would be the Michaelis–Menten model (often used for modelling the rate of enzymatic reactions), which can be written

$E(Y|x) = \frac{\lambda}{1+\kappa/x} = \frac{\lambda\, x}{\kappa+x}$

where $Y$ is typically a random variable representing a reaction rate and $x$ is the concentration of a substrate; $\lambda$ and $\kappa$ are parameters (as is the variance of the implied noise term not shown here) representing the maximum rate at saturation and the substrate concentration required to achieve half the maximum rate respectively. Even though it's a parameter defining the conditional mean of $Y$ (i.e. defining the "regression"), the $\kappa$ parameter certainly doesn't enter this regression model linearly.

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